An existence result for nonlinear elliptic equations on $\Bbb R^d$ without sign condition

More by Yasuhiro Fujita

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Abstract

In this paper, we consider the nonlinear elliptic equation of the following type:
$$
-\frac{1}{2} \Delta u(x) + \nabla w(x)\cdot \nabla u(x) +
[\lambda + H(x,u(x))]u(x) = f(x),\qquad x \in {\mathbb R}^d,
$$
where $\lambda$ is a given constant and $f$, $H$, and $w$ are
given functions, respectively.
The derivative $\nabla w$ of the function $w$ is unbounded on
${\mathbb R}^d$.
Our purpose is to show the existence of a solution to this
equation without sign conditions on $H$.
Therefore, we can treat even the case that $H$ is unbounded
below on ${\mathbb R}^d$.
This is due to the existence of the term $\nabla w\cdot \nabla u$.